1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. THE COMPACTNESS OF /';;-SOLUTIONS 145

for (y,t) E B 9 k(- 2 r~) (zk,rk) x [-r~,O]. Using IRc-gkl:::; R9k, we obtain that

fort E [-r~, OJ,

---at 8R-tik ( Zk, t ) :::; l:!.R 9 k ( zk, t) + 2R^2

9 k (zk, t)

:::; C (C1, n) REk (zk, 0)

for some constant C ( C1, n) < oo depending only on C 1 and n. Choosing

. { 1 1}

a=mm 2C(C1,n)'2 <l,

we have

8 R 9 k ( ) 1 2 1 _ 4

---af Zk, t :::; 2 a R 9 k (zk, 0) = 2 a rk

fort E [-r~, OJ. Integrating the above equation from -ar~ to 0, we obtain

1 2

(20.39) 2R9k (zk, 0) :::; R9k (zk, -ark)·

Combining (20.37) and (20.39), we get the following uniform bound at

(zk, 0):

(20.40)

Now by combining Steps 2 and 3 we obtain the following.

STEP 4. Curvature bounds in balls centered at Xk·

Specifically, as a consequence of combining (20.31) and (20.40), we con-

clude that there exists C3 > 0 independent of k such that

R-gk (y, 0) :S; C'3^2 for y E B-gk(o) (xk, C3).

Hence the volume lower bound

(20.41)

follows from the 11:-noncollapsing assumption.

STEP 5. Bounds of the curvature, independent of k, in balls centered at

Xk with arbitrarily large radii.

Given any r > 0, for any y E B-gk(o) (xk, r) we have

B9k(o) (xk, C3) C B9k(o) (y, r + C3)

and hence
Vol-gk(o) B-gk(o) (y, r + C3) 11:C'£'
(r + C3t :::::: (r + C3r·

We shall show that there exists a constant C4 < oo, depending on rand

on the sequence but independent of k, such that

(20.42) R9k (y, 0) · (r + C3)2 ::S: C4